Tree Spanners for Bipartite Graphs and Probe Interval Graphs

Tree Spanners for Bipartite Graphs and

Probe Interval Graphs

Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Ueha

ra3

1 Universität Rostock

2 Kent State University

3 Komazawa University

Tree Spanners for Bipartite Graphs and

Probe Interval Graphs

Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Ueha

ra3

1 Universität Rostock

2 Kent State University

3 Komazawa University

Tree Spanner

Spanning tree T is a tree t-spanner iff

dT (x,y) ≦t dG (x,y)

for all x and y in V.

G T

xy

xy

Tree Spanner

Spanning tree T is a tree t-spanner iff

G T

dT (x,y) ≦ t dG (x,y)for all {x,y} in E.

Tree Spanner

Spanning tree T is a tree 6-spanner.

G T

Tree Spanner

G admits a tree 4-spanner (which is optimal). Tree t-spanner problem asks

if G admits a tree t-spanner for given t.

G T

Applications in distributed systems and communication networks

synchronizers in parallel systems topology for message routing

there is a very good algorithm for routing in trees

in biology evolutionary tree reconstruction

in approximation algorithms approximating the bandwidth of graphs

Any problem related to distances can be solved approximately on a complex graph if it admits a good tree spanner

G

7-spanner for G

Known Results for tree t -spanner general graphs [Cai&Corneil’95]

a linear time algorithm for t =2 (t=1 is trivial) tree t -spanner is NP-complete for any t 4≧　　 ( NP-completeness of ⇒ bipartite graphs for t 5)≧ tree t -spanner is Open for t=3

Known Results for tree t -spanner chordal graphs [Brandstädt, Dragan, Le & Le ’02]

tree t -spanner is NP-complete for any t 4≧ tree 3-spanner admissible graphs [a Number of Authors]

cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs

tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99]

tree 3 -spanner is in P for planar graphs [FK’2001]

Known Results for tree t -spanner chordal graphs [Brandstädt, Dragan, Le & Le ’02]



tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99]

tree 3 -spanner is in P for planar graphs [FK’2001]

⇒ Bipartite Graphs??

Known Results for tree t -spanner bipartite graphs [Cai&Corneil ’95] tree t -spanner is NP-complete for any t 5≧ chordal graphs [Brandstädt, Dragan, Le & Le ’02]



convex bipartite interval bigraphs ⊂ ⊂ 　 bipartite ATE-free graphs ⊂ chordal bipartite graphs ⊂ bipartite graphs

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

chordalbipartite

intervalbigraph

convex

AT-free bipartiteATE-free

bipartite

NP-C

4-Adm.

3-Adm.

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

enhancedprobe

interval

chordalbipartite

probeinterval

intervalbigraph

convexSTS-probe

interval


bipartite

NP-C

4-Adm.

3-Adm.

=

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

enhancedprobe

interval

chordalbipartite

probeinterval

intervalbigraph

convexSTS-probe

interval


bipartite

NP-C

4-Adm.

3-Adm.

=

7-Adm.

NP-hardness for chordal bipartite graphs

[Thm] For any t 5, the ≧ tree t-spanner problem is NP-complete for chordal bipartite graphs.

Reduction from 3SATMonotone

… (x, y, z) or (x, y, z)


Reduction from 3SAT Basic gadgets

Monotone

… (x, y, z) or (x, y ,z)

S1[a,b] S2[a,b] S3[a,b]

a

a’

b

b’

a b

a’ b’

S1[a,a’]

S1[a’,b’]

S1[b,b’] S2[a,a’]

S2[a’,b’]

S2[b,b’]

a b

a’ b’


Reduction from 3SAT Basic gadget Sk[a,b] and its spanning trees

Monotone

… (x, y, z) or (x, y ,z)

a

a’

b

b’

a

a’

b

b’a

a’

b

b’

H

with {a,b}

(2k+1)-spanner

without {a,b}

h

(2k+h)-spanner

a

a’

b

b’

without {a,b}

(2k-1)-spanner


Reduction from 3SAT Gadget for xi

Monotone

… (x, y, z) or (x, y ,z)

q r

sp

xi xi

xixi

xi

xi1

2 m1

2 m…

…Sk-1[]

Sk[]× ２

＝

＝

Must be selected


Reduction from 3SAT Gadget for Cj

Monotone

… (x, y, z) or (x, y ,z)

cj cj

djdj

Sk[]× ２＝

+ -

+ -


Reduction from 3SAT Gadget for C1=(x1,x2,x3) and C2=(x1,x2,x4)

Monotone

… (x, y, z) or (x, y ,z)

q r

sp

x1 x1

x1x11

21

2

Sk-2[]＝

x2 x2

x2x21

21

2

x3 x3

x3x31

21

2

x4 x4

x4x41

21

2

c1 c1

d1d1

+ -

+ -c2 c2

d2d2

+ -

+ -

Tree 3-spanner for a bipartite ATE-free graph

An ATE(Asteroidal-Triple-Edge) e1,e2,e3 [Mul97]:Any two of them there is a path from

one to the other avoids the neighborhood of the third one.

[Lamma] interval bigraphs biparti⊂te ATE-free graphs chordal bi⊂partite graphs.

e1

e3e2


A maximum neighbor w of u: N(N(u))=N(w)

[Lamma] Any chordal bipartite graph has a vertex with a maximum neighbor.

u w

chordal bipartite graph⇔•bipartite graph•any cycle of length at least 6 has a chord


G; connected bipartite ATE-free graph u; a vertex with maximum neighbor

For any connected component S induced by V ＼ Dk-1(u), there is w in Nk-1(u) s.t. N(w) S∩N⊇

k(u)

Su … w


Construction of a tree 3-spanner of G: u; a vertex with maximum neighbor

u … w

Conclusion and open problems• Many questions remain still open. Among them:

• Can Tree 3–Spanner be decided efficientlyon general graphs??? on chordal graphs?on chordal bipartite graphs?

•Tree t–Spanner on (enhanced) probe interval graphs for t<7?

Thank you!

Documents

Tree Spanners for Bipartite Graphs and Probe Interval Graphs